3.420 \(\int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx\)

Optimal. Leaf size=48 \[ \frac {\sqrt {\frac {2}{5}} \tanh ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )+1}}\right )}{e} \]

[Out]

1/5*arctanh(1/2*sin(d+e*x-arctan(3/4))*2^(1/2)/(1+cos(d+e*x-arctan(3/4)))^(1/2))*10^(1/2)/e

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Rubi [A]  time = 0.06, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3115, 2649, 206} \[ \frac {\sqrt {\frac {2}{5}} \tanh ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )+1}}\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]

[Out]

(Sqrt[2/5]*ArcTanh[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[1 + Cos[d + e*x - ArcTan[3/4]]])])/e

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3115

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Int[1/Sqrt[a +
Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx &=\int \frac {1}{\sqrt {5+5 \cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}} \, dx\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{10-x^2} \, dx,x,-\frac {5 \sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {5+5 \cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}}\right )}{e}\\ &=\frac {\sqrt {\frac {2}{5}} \tanh ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {1+\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}}\right )}{e}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 101, normalized size = 2.10 \[ -\frac {\left (\frac {2}{5}+\frac {6 i}{5}\right ) \sqrt {\frac {4}{5}+\frac {3 i}{5}} \tan ^{-1}\left (\left (\frac {1}{10}+\frac {3 i}{10}\right ) \sqrt {\frac {4}{5}+\frac {3 i}{5}} \left (3 \tan \left (\frac {1}{4} (d+e x)\right )-1\right )\right ) \left (\sin \left (\frac {1}{2} (d+e x)\right )+3 \cos \left (\frac {1}{2} (d+e x)\right )\right )}{e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]

[Out]

((-2/5 - (6*I)/5)*Sqrt[4/5 + (3*I)/5]*ArcTan[(1/10 + (3*I)/10)*Sqrt[4/5 + (3*I)/5]*(-1 + 3*Tan[(d + e*x)/4])]*
(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2]))/(e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])

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fricas [B]  time = 1.17, size = 147, normalized size = 3.06 \[ \frac {\sqrt {5} \sqrt {2} \log \left (-\frac {9 \, \cos \left (e x + d\right )^{2} + {\left (13 \, \cos \left (e x + d\right ) - 6\right )} \sin \left (e x + d\right ) + 2 \, {\left (\sqrt {5} \sqrt {2} \cos \left (e x + d\right ) - 3 \, \sqrt {5} \sqrt {2} \sin \left (e x + d\right ) + \sqrt {5} \sqrt {2}\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5} - 33 \, \cos \left (e x + d\right ) - 42}{9 \, \cos \left (e x + d\right )^{2} + {\left (13 \, \cos \left (e x + d\right ) + 14\right )} \sin \left (e x + d\right ) + 27 \, \cos \left (e x + d\right ) + 18}\right )}{10 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="fricas")

[Out]

1/10*sqrt(5)*sqrt(2)*log(-(9*cos(e*x + d)^2 + (13*cos(e*x + d) - 6)*sin(e*x + d) + 2*(sqrt(5)*sqrt(2)*cos(e*x
+ d) - 3*sqrt(5)*sqrt(2)*sin(e*x + d) + sqrt(5)*sqrt(2))*sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5) - 33*cos(e*
x + d) - 42)/(9*cos(e*x + d)^2 + (13*cos(e*x + d) + 14)*sin(e*x + d) + 27*cos(e*x + d) + 18))/e

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5), x)

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maple [A]  time = 0.25, size = 77, normalized size = 1.60 \[ -\frac {\left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+5}\, \sqrt {10}\, \arctanh \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+5}\, \sqrt {10}}{10}\right )}{5 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x)

[Out]

-1/5*(1+sin(e*x+d+arctan(4/3)))*(-5*sin(e*x+d+arctan(4/3))+5)^(1/2)*10^(1/2)*arctanh(1/10*(-5*sin(e*x+d+arctan
(4/3))+5)^(1/2)*10^(1/2))/cos(e*x+d+arctan(4/3))/(5+5*sin(e*x+d+arctan(4/3)))^(1/2)/e

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )+5}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*cos(d + e*x) + 3*sin(d + e*x) + 5)^(1/2),x)

[Out]

int(1/(4*cos(d + e*x) + 3*sin(d + e*x) + 5)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} + 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+4*cos(e*x+d)+3*sin(e*x+d))**(1/2),x)

[Out]

Integral(1/sqrt(3*sin(d + e*x) + 4*cos(d + e*x) + 5), x)

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